Differential and Integral Equations

Liouville theorems for integral systems related to fractional Lane-Emden systems in $\mathbb{R}^N_+$

Senping Luo and Wenming Zou

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In this paper, we consider some integral systems in the half space $\mathbb R^N_+$ and obtain Liouville type theorems about the positive solutions. By moving plane method in terms of the integral form, we shall see that the positive solution $(u(x_1,...,x_N), v(x_1,...,x_N))$ of the integral systems must be independent of the first $(N-1)$-variables, i.e., $u=u(x_N),v=v(x_N)$. Then, combine with the order estimates about $x_N$, we reduce the problem to a sequence of algebraic systems. Furthermore, we discuss the relationship between the integral system and the fractional differential system related to the fractional Lane-Emden equations. By this way, we obtain two non-existence theorems for the fractional differential system.

Article information

Differential Integral Equations, Volume 29, Number 11/12 (2016), 1107-1138.

First available in Project Euclid: 13 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65R20: Integral equations 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35R11: Fractional partial differential equations


Luo, Senping; Zou, Wenming. Liouville theorems for integral systems related to fractional Lane-Emden systems in $\mathbb{R}^N_+$. Differential Integral Equations 29 (2016), no. 11/12, 1107--1138. https://projecteuclid.org/euclid.die/1476369332

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